标题： 物质的多拓扑相 显示英文标题

标题： Multi-topological phases of matter

摘要： 物质的拓扑相和拓扑边界态的发现对凝聚态物理和光子学产生了巨大影响，其中拓扑相通过能带定义，从而产生拓扑能带理论。然而，无法通过能带拓扑描述但仍然支持非平凡边界态的拓扑系统鲜为人知且研究甚少。在此，我们揭示了一种新的物质拓扑相，称为“多拓扑相”（MTP），其特征是具有多组边界态，每组边界态与一个不同的拓扑不变量相关。与传统的拓扑相变不同，MTP相变可以在不关闭能隙的情况下发生。我们在一维拓扑绝缘体和二维高阶拓扑绝缘体中展示了MTP的典型例子，根据能带拓扑，这些系统通常是平凡的。此外，MTP也可以存在于间接能隙陈绝缘体中，超出传统体-边界对应关系预测边界态存在的范围。实验上，我们在激光写入的光子晶格中演示了MTP的前两个例子。我们的发现是拓扑物理领域的基础性进展，并为设计新型拓扑材料提供了途径。 显示英文摘要

摘要： The discovery of topological phases of matter and topological boundary states had tremendous impact on condensed matter physics and photonics, where topological phases are defined via energy bands, giving rise to topological band theory. However, topological systems that cannot be described by band topology but still support non-trivial boundary states are little-known and largely unexplored. Here, we uncover a new kind of topological phase of matter named "multi-topological phase" (MTP) that features multiple sets of boundary states, where each set is associated with one distinct topological invariant. Unlike conventional topological phase transitions, the MTP transitions can occur without band-gap closing. We present typical examples of MTPs in a one-dimensional topological insulator and a two-dimensional higher-order topological insulator, where the systems are otherwise trivial according to band topology. Furthermore, MTPs can exist also in indirectly gapped Chern insulators, beyond the regime where the conventional bulk-boundary correspondence predicts the existence of boundary states. Experimentally, we demonstrate the first two examples of MTPs in laser-written photonic lattices. Our findings constitute a fundamental advance in topological physics and provide a route for designing novel topological materials.